a-theater-owner-knows-that-to-make-a-profit-as-well-as-to-comply-with-fire-regulations-the-number-of

Name: a theater owner knows that to make a profit as well as to comply with fire regulations the number of
Uploaded: 2019-08-07
Duration: 3 min 4 s
Channel: Numerade
Description: A theater owner knows that to make a profit as well as to comply with fire regulations, the number of tickets that he sells can differ from 225 by no more than $75 .$ How many tickets can the theater owner sell in order to make a profit and comply with fire regulations?

Question

A theater owner knows that to make a profit as well as to comply with fire regulations, the number of tickets that he sells can differ from 225 by no more than $75 .$ How many tickets can the theater owner sell in order to make a profit and comply with fire regulations?

Joshua Utley · Accepted Answer

Okay, so we have a theater owner that needs to comply with fire regulations. And it says that that the difference between the number of tickets he sells and 2 25 can never be more than 75. And what that means is, in order to sell a prophet he need, he can&#x27;t sell 75 less than 2 25 But in order to comply with fire regulations, he can only sell up to 75 more than 2 25 because otherwise the theater would get too crowded. So what this is saying is that the distance from 75 from 2 25 is it is. It&#x27;s an absolute value. So the distance between 2 25 and the amount of tickets I sell has to be less than or equal to 75 and I want to make sure it&#x27;s less than or equal to 75. It is, and so that&#x27;s the absolute value when you say said, and now it wants to know how many tickets does the theater owner have to sell in order to make a profit and at the same time comply with fire regulations? And so now we just need to solve for the number of tickets here, which is X in this case. And so we split our equation, since the absolute values already isolated to 25 minus X is less than or equal to 75 2 25 minus X is greater than or equal to minus 75. And so to solve this one, we just subtract 2 25 75 minus 2 25 is minus 1 50 So x is less than or equal to minus 1 50 Here we divide by minus on. And so we have to flip our sign. So we have X is bigger than 1 50 and we did the same thing here. Subtract 2 25 over. And so we&#x27;re gonna have minus X is greater than our equal to minus 300. And then we divide by negative one, which means we have to flip our sign. So we have X is less than or equal to 300. And so what he needs to do is he needs to have all of this ticket sales be in the range of 1 50 2 300 with 2 25 being in the middle. And that should make sense because what&#x27;s 75 away from 2 25? 300? So he can go that far away. Up 300. He&#x27;s just barely meeting his fire regulations, his theaters just about to be overcrowded. But if he only sells 1 50 then he&#x27;s probably breaking even, or just barely making a profit. And so there&#x27;s this range of 150 numbers of tickets take itself, and this is the range where the theater is making a profit.

Brandon Fox · Answer

Let&#x27;s take our answer number 2500 divided by $4. Try to find out much tickets must be sold. So four goes into 25. Um, let&#x27;s say six times. That&#x27;s 24. It&#x27;s a track to get one drop of zero four goes into 10 twice on that&#x27;s in the AIDS of track it to and dropped zero in four goes into 25 times. Four times five is 20 track and get zero, and our answer is 625 tickets.

Sinisa Stura · Answer

Hello. Hello. This problem is about a musical theory group Theatre group that when the price is it does they can feel it. Receipts in the theater, which is all that survival units and for every additional dollar charged, the number of people buying tickets decreases when the changing prices plus $1 the change in the tickets He&#x27;s monitoring device. What ticket price. So if we express driven you as a function of being what the compress maximizes revenue, we want the people we want the p for rich, the revenue he&#x27;s makes my says several things are employed here. We need to the ruin. Your function is Pete. I&#x27;m secure. So we need to represent Q as functional feet. Okay, given a linear connection between pink ears, uh, once again when we have a linear function, why? Well, why is there any a function of X? Then it&#x27;s forming something like this and am is calculated as the changing. Why over the change makes so here we will have, uh, que as faction of p. So our slip will be ah, changing Q over the change in. We must look after living here for a change in Hughes minus 75 over one, which is minus sense. But we will also employed the point slope form of a line which is giving us. And this gives us one point pick you on the line is eight 1500 if we go. So first we find Q as a functional, which we will then insert here to find our as functional. The Q B He&#x27;s a thing by Q minus. Q. One Thank you months 100 equals. Aim is my 75 minus 75 p. It&#x27;s a Q equals amount of 35 be below US 600 and when we transfer this to the other side another 1500. Alright, So kid pass affection of bees my 75 b, the last 2100 we do this. We now find our as a function of being food are well are as a function of being is a thing by multiple ng be with cube, especially bees minus 35. The blessed Basil 100 which is minus 75 p squared class 2100 being so, the function are being ese the This is in fact, a quadratic formula, a quadratic pardon, affection and it&#x27;s a graph is a proble. Since the leading coefficient is negative, it&#x27;s going to be opening them and the extreme point or the verdicts. He&#x27;s going to be a maximum points. Finally, maximum points. We need to find the derivative of our by. There it will be so that their and equated zero We do that here. So there is. This term is two times when it&#x27;s well 75 which is what minus 50 p pull up to those 100 equal zero we&#x27;ve got to the other side 2100 equals 150 p divided by 1 50 50 Ah Spee This is going to be four thing, But let&#x27;s trick with a calculator it is 14 and that the fact is all that the question asks the question asked, What is the price at which revenue is maximum good press that they&#x27;re looking for us? $14 per tickets? Let&#x27;s see. Do they need the actual revenue? No, they don&#x27;t. So we don&#x27;t There we go. Hope this helps. Heavens, they Bye.

Danny Huang · Answer

Okay, so for problem 106 We have tickets to the Shakespeare Community Theater at $10. So tickets $10. Um, and then the average attendance is 25 people, 250 people. And then if we decrease the price by 50 cents, 25 more people attend. So the first question is, ah, what ticket price to maximize Rather new. And then once you find that price, what&#x27;s the average attendance? Okay, so first revenue is equal to price a ticket times people attending. So that&#x27;s right T for ticket price and p for people attending. So we know that the ticket price is related to that number of people attending in that, um, that number of people attending is equal to 250. All right, so let&#x27;s graphic firs. Right? So when ticket price is 10 I take it ti ticket prices 10. We have 250 people. Let&#x27;s make that a little bit lower. So 2 50 10 and then for each 0.5 decrease in ticket price, 25 more people, 10. So if we were to go. So this is like a linear relationship that way. And if we were to go all the way back to making it completely free. That would be 10 times, um, 10 times, too. We&#x27;re wait, they&#x27;ll be so 10. Did I have my point? Five is going to be 20 and then 20 times the 25 people for each A 50% decrease is going to be equal to 500 without the year 500 people difference. There&#x27;ll be 7 50 Right quick relationship is number of people attending. It&#x27;s gonna start at 7 50 and then it&#x27;s going to be subtracting 0.5, um, or for each dollar, increase is going to be 50 people. Last will 10 so minus 50 t All right, so now we have the relationship. And then so we have our before that revenue is equal to tee times p. Um And then we already have peace so we can solve for P helping t time 7 50 minus 50 t, which is equal to 7 50 t minus 50 t squared and then worried to maximize revenue. We want to find when the directive is equal to zero. So the rt T is equal to zero, which is equal to So we used the powerful for both of these. And now give us two times 50 is 100. And that&#x27;s track one from exponents. And now we just saw for tea, which gives us 7 50 is equal to 100 t or the ticket price should be 7 50 over 100 which is seven point five dollars. Um, so that&#x27;s the ticket prices will maximize the theaters revenue. And then now we want to find the average attendance at that price. And so we&#x27;ll just go back to the relationship we found. Here we have p equals 7 50 minus 50 times 7.5. There&#x27;s gonna be some 50 minus 3 75 which is equal to 3 75 people. That&#x27;s the average attendance.

Ankit Gupta · Answer

let the number off adult patrons B X so therefore number off child patrons or Children will be 5200 minus six. Then, according tothe the question, our equation will become 7.50 x plus 4.50 multiplied with 5200 minus X is equal to 29,961. So now simply find will get 7.5 X plus that is traded 3400 minus 4.5. X is equal to 29,961 now rearranging. To keep the light terms together, we&#x27;ll get three. X is equal to 29,961 minus 23,400 which gives US three. X is equal to 6561 6561. So now dividing both sides by three, we get X as equal toe to 187 So therefore, the number off adult patrons is 2000 100 on 87

Charles Carter · Answer

Okay, So each ticket costs $30 minus 25 cents. Uh, her tickets sold. And so if we multiply that by the number of tickets sold, we can get our profit from tickets when we take away $200. And that gives us our profit. So, in order, Teoh, find what the best number tickets to sell to get the most amount of profit. I was gonna take the derivative of this, which is just gonna be 30 minus 0.5 x. They were set that equal to zero and said X equals 60. So this companies to sell 60 tickets to maximize their profit. Thank you, very.

Problem

A cereal bar is listed as containing 200 calories…

Problem 28 Hard Difficulty

A theater owner knows that to make a profit as well as to comply with fire regulations, the number of tickets that he sells can differ from 225 by no more than $75 .$ How many tickets can the theater owner sell in order to make a profit and comply with fire regulations?

Answer

Between 150 and 300 tickets inclusive

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Ann Xavier Gantert

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Alayna H.

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Absolute Value - Example 1

Absolute Value - Example 2

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Between 150 and 300 tickets inclusive

Algebra 2 and Trigonometry

Grace H.

Alayna H.

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Absolute Value - Example 1

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Ann Xavier GantertAlgebra 2 and Trigonometry

Grace H.

Alayna H.

Caleb E.

Kristen K.

Ann Xavier Gantert

Algebra 2 and Trigonometry